Accelerated Optimized Topology Design in Affine Formation Control Using ADMM

This paper studies the problem of topology design for activating affine formation control schemes. The affine formation control exhibits its unique feature as it relies on the stress matrix to dynamically maneuver the whole formation by controlling a small number of agents. Network properties of interest for this design problem generally give rise to optimization formulations within the framework of mixed-integer semidefinite programming (MISDP), resulting in computational inefficiency and NP-hardness. Firstly, to avoid introducing binary variables, the optimization of communication cost is modeled as an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_{1}$</tex-math></inline-formula>-regularized network sparsity problem. In this way, an optimized topology design method accelerated by the alternating direction method of multipliers (ADMM) is proposed to obtain the stress matrix with low communication cost, fast convergence speed and high tolerance to time-delay. Furthermore, addressing scenarios irrespective of whether the minimum eigenvalue of the stress matrix is prescribed, we propose two enhanced ADMM-based algorithms with closed-form solutions. This is achieved through the transformation of semi-definite constraints in the subproblem into equality constraints. Finally, comparative simulations demonstrate the accelerated effects of the proposed scheme, showcasing its effectiveness in interaction topology construction and optimization for large-scale networks.